What causes double rainbows? Why is the order of the colors reversed?

While on vacation recently our family
saw a double rainbow. How do these occur? Why did the second faint
rainbow (which was on top of the first) have the opposite color
sequence?

Ah, rainbows – those will-o'-the-wisps of ethereal beauty, made
possible only by a full lineup of optics phenomena. Let's run down the
list:

1. Refraction. When a ray of sunlight strikes a raindrop, the
ray refracts, or bends, at the point where it passes out of the air and
into the water of the drop. The angle of the bend is determined by (a)
the intrinsic light-transmitting properties of air and water (every
transparent substance has its own individual index of refraction) and
(b) the angle at which the ray strikes the surface of the spherical
droplet – whether, e.g., it hits the drop squarely or strikes a glancing
blow off to one side.

2. Dispersion. Meanwhile, the drop is acting as a prism,
splitting the white light of the ray into its component colors by
refracting the different wavelengths at different angles: red
wavelengths bend a certain amount, orange wavelengths a slightly
different amount, and so on.

3. Internal reflection. Most of the light striking the raindrop
passes straight through it and out the far side, but some of it
reflects off the rear interior surface of the drop and is sent in some
new direction. The ratio of light transmitted to light reflected is,
once again, a function of the angle at which the ray hits the surface.

4. Refraction and dispersion, part 2. When the reflected light exits the drop and re-enters the air, it's refracted and dispersed a second time.

Light rays emitted by the sun are effectively parallel when they reach
the earth, and raindrops are effectively all the same shape. So when
sunlight shines into a sky full of raindrops, it's encountering millions
of tiny, very similar spherical prisms and interacting with each in
pretty much the same way: each produces a basically identical pattern of
refracted, dispersed, reflected, and re-refracted light in a spectrum
of colors. The reflected red light is at its greatest intensity at an
angle of 42 degrees from the direction of the sun's rays, while the
violet light has maximum intensity at 40 degrees. When you face a rainy
sky with the sun at your back you see a ring of red light, forming the
outer edge of the rainbow, at 42 degrees from the direction of the
sunlight, a violet ring at 40 degrees forming its inner edge, and all
the other colors of the spectrum in between. The rainbow is entirely an
optical illusion; it changes its apparent position in the sky as you
change your vantage point, meaning that no two people are ever seeing a
rainbow the same way (and explaining why that pot of gold is so
elusive). Also, because the light forming the rainbow is reflected at
angles of 40 to 42 degrees, for the most part rainbows are seen only
during the hours around sunrise and sunset: if the sun is higher than 42
degrees in the sky the rainbow reflected by the raindrops will be below
the horizon for an observer at ground level. You get better viewing at
greater altitude, and it's possible to see complete circular rainbows
from an airplane.

Now, about double rainbows: What's happening here is that the ray of
sunlight bounces twice off the back interior surface of the raindrop
before re-emerging into the air. The second reflection inverts the order
of the colors – the secondary violet band forms at 54 degrees, the red
band at 50.5 degrees – so the secondary rainbow appears above the
primary one, with red on the inner edge and violet on the outer. Because
the twice-reflected light has had two chances to be transmitted out the
back of the raindrop rather than reflected back toward the observer,
the secondary bow is much fainter than the primary and frequently cannot
be seen at all; it's typical for a secondary rainbow to be visible only
at certain points along the arc.

If the light is strong enough to remain visible after being reflected three
times inside the raindrop, an even fainter tertiary rainbow can
sometimes be seen (at least in part) above the secondary one, with the
red back on the outside and the violet on the inside. And rumor has it
that it's occasionally possible to see a quadruple rainbow.

Nitpickers will ask: What about diffraction? Doesn't it play a role here
too? All I have to say is (a) yes, diffraction – a quantum phenomenon
where light waves cancel each other out or amplify one another –
sometimes figures in rainbow formation, if the raindrops are small
enough, in which case (b) all bets are off – you might get smaller
rainbows inside the main bow, you might get rainbows with the red in the
middle – but (c) no way am I going to work out the math for this. If
you're desperate to know this kind of stuff, well, that's why they
invented physics grad programs.